For consensus reaching process in a social network, the leader may influence the other individual experts, called followers. In this section, we study the followers to adjust their decision matrices according to the leader’s matrix in order to reach a consensus.
3.1. Problem Formulation
For the sake of convenience, the multi-criteria Pythagorean fuzzy group decision making problem in this paper is formulated as follows.
Let () be a collection of feasible alternatives, be a set of criteria, and be the weight vector of the criteria, which satisfies and . Let be a group of experts with the adjacency matrix , and be the weight vector of the experts, where and . Suppose that the expert evaluates the alternative () with criterion () by PFN . Moreover, for the PFN , shows the degree to which the alternative satisfies the criterion and shows the degree to which alternative dissatisfies the criterion . Then, the decision matrix of the expert can be denoted as ().
3.2. The Leader-Following Consensus Reaching Method
The similarity degree between any two experts can be calculated as follows.
Definition 6. Letandbe two experts with Pythagorean fuzzy group decision matricesand, respectively. The similarity degreebetweenand is defined as follows:
Theorem 1. For expertsand, whose similarity degrees are shown in Equation (6),andremain constant.
Proof of Theorem 1. According to Equation (2), we have
Thus, can be obtained and can be easily proven. □
In this paper, we define
as the similarity matrix of all the experts where
. For any two directly connected experts in a social network, the similarity degree can be taken as the direct connection strength. Therefore, the undirected social network becomes a weighted one, and the corresponding adjacency matrix is denoted by
, where
Moreover, we can obtain by , where the operator is the Hadamard multiplication of the matrix.
For the two experts who are indirectly connected in a social network, we can use the Einstein product to evaluate their connection strength [
24].
Definition 7. Letand be two indirectly connected experts is a social network, where the shortest path from
to is . The connection strengths for , , …, are denoted as , , …, , respectively. Then, the indirect connection strength from to can be obtained by According to Equation (8), inequality
always remains constant [
24]; that is, the connection strength between two indirectly connected experts is not more than any connection strength between the two experts and the intermediary expert, which is common sense. If there is more than one path between them, then only the shortest indirect path is used. Moreover, more than one shortest path exists, the average connection strengths of the shortest paths are computed.
All the connection strengths between any two experts can be obtained according to Equations (6)–(8). Then, we can construct a connection strength matrix
, where
Moreover, if an expert has a stronger connection strength than the other experts, then we can think that they are more important. Thus, we can obtain the comprehensive weight vector
associated with the original weight vector as follows:
In a social network, if a node (expert) has the most significant connection strength compared with other nodes, then it can be seen as the leader in this social network.
Definition 8. For any two nodes (experts)andin an undirected social network withnodes, the connection strength is, where. Then, the leaderof thenodescan be identified by If the leader has been identified in a social network, then the other nodes (experts) are likely to follow the leader’s decision information in the consensus reaching process. When the followers’ decision matrices are close to the leader’s decision matrix in a certain range, it can be considered that the nodes (experts) have reached a satisfied consensus. In this case, the consensus index, called the leader–following consensus index , can be defined as follows.
Definition 9. In an undirected social network withnodes, their corresponding normalized decision matrices are, and the leaderwith its decision matrixis identified by Equation (11). The leader-following consensus index of this group of nodesis defined bywhere.
For a given threshold value
, if
holds, then the group reaches a satisfied consensus. Otherwise, some followers’ decision matrices should be adjusted to reach a satisfied consensus using the following Algorithm 1.
Algorithm 1. Adjustment for Satisfied Leader-Following Consensus Reaching Index. |
Inputs: The leader and its decision matrix , the normalized decision matrices , the maximum number of iterations , and the threshold value . |
Outputs: Adjusted decision matrices , the iteration step , and satisfied leader-following consensus index . |
Step 1: Set and , . Step 2: Calculate the similarity degree between each follower and the leader by
where
If or , then go to Step 4. Otherwise, go to Step 3. Step 3: Suppose that . Then, let:
Set and go to Step 2. Step 4: Let () and . Output , , and the number of iterations . Step 5: End.
|
A survey of the literature shows that the threshold value
is often subjectively determined by the expert(s), as in this group, or by a super expert [
25]. The determination of the threshold value
may be seen as an extra burden on the group. However, it does have some advantages as it provides a viable option for the group to control the decision process schedule [
26].
Theorem 2. Letbe the leader-following consensus index in theiteration of Algorithm 1 andbe the index in theiteration. Then, the following inequality holds. Please see the
Appendix A for the proof of Theorem 2.
Theorem 2 can guarantee that the leader-following consensus index increases with each iteration by performing Algorithm 1. Moreover, for a given threshold value , if at the beginning, the group of experts must reach a satisfied leader-following consensus using Algorithm 1 with enough iterations and a suitable control parameter . The above result can be shown as the following theorem.
Theorem 3. Letbe a given threshold value. Ifbefore performing Algorithm 1, then after sufficient iterations, the new leader-following consensus indexmust reach a satisfied consensus, meaning the following inequality holds Similarly, for the other experts’ decision matrices, where their similarity degrees are less than the threshold value, the corresponding similarity degrees can reach the given threshold value using Algorithm 1. Thus, the minimum similarity degree to the leader reaches the given threshold (i.e., Theorem 3 is proven).
It is worth noting that the social network is stable in the group decision making process, we adjust the decision matrix to reach the satisfied consensus level using Algorithm 1.